[math-fun] Draft of my September 2016 blog post
I started writing a new draft titled "Going Negative" and would love to get your feedback. I plan on publishing it on the 17th and will continue to tinker with it between now and the weekend. Is it too long? If so, what passages would you suggest I compress, or cut? I'm not an historian. Do I get my math history wrong, either blatantly (with mis-statements of fact) or subtly (by misplacement of emphasis or other forms of distortion)? I'm still confused about the differences between Frend's position and De Morgan's (and I'm sure the difference is an important one). I'm not a pre-college teacher. Do I misunderstand the pedagogical issues posed by negative numbers at the pre-college level? Suggestions for references, and comments of all kinds, are welcome. I appreciate candid criticism from people who are sympathetic to my aims but think I've fallen short of them. My prose style is designed to make the material accessible, but my breeziness should not be interpreted as indicating indifference to scholarly correctness. Please leave your feedback at https://mathenchant.wordpress.com?p=1196&shareadraft=57d777dd7094f keeping in mind that the comments of EVERYONE on math-fun will get funneled through the same email conduit; if you want me to know who you are, please sign your comments! If you sign your comments I'll assume (unless you indicate otherwise) that you don't mind my acknowledging your contribution. Title: Going Negative Beginning: Minus times minus equals plus / The reason for this we will not discuss. --- W. H. Auden, recalling a popular verse from his school days Ever tried mixing together your two least favorite foods? I suspect you haven't. Nobody mixes two noxious ingredients and expects the results to be tasty... Read more: https://mathenchant.wordpress.com?p=1196&shareadraft=57d777d d7094f Thanks, Jim Propp
My understanding is that in earlier versions of the English language (e.g., Middle English), the idea of a "double negative", as we know it today, didn't exist. Rather, the more negatives one threw into a sentence, the stronger the sense of the negation. So saying "I don't not like that" meant you *really* didn't like that, as opposed to the modern double negative interpretation of "I don't (not like that)", i.e. I don't dislike that, which I would take to mean I'm either neutral or positive towards it. I suspect mathematical logic, and multiplication of two negatives producing a positive, may have influenced the English language in this regard. Conversely, the old interpretation of double negatives in the English language may have prejudiced people against the notion of the product of two negatives being positive.
From a purely mathematical standpoint, having the product of two negatives be positives breaks some basic identities. For instance, (10 - 5) * -10 is clearly -50. But if the distributive law applies to negative numbers, then (10 * -10) + (-5 * -10) should also be 50, assuming that (10 - 5) is the same as (10 + -5). This works if (-5 * -10) is 50, but fails horribly if it's -50.
Tom James Propp writes:
I started writing a new draft titled "Going Negative" and would love to get your feedback. I plan on publishing it on the 17th and will continue to tinker with it between now and the weekend.
Is it too long? If so, what passages would you suggest I compress, or cut?
I'm not an historian. Do I get my math history wrong, either blatantly (with mis-statements of fact) or subtly (by misplacement of emphasis or other forms of distortion)? I'm still confused about the differences between Frend's position and De Morgan's (and I'm sure the difference is an important one).
I'm not a pre-college teacher. Do I misunderstand the pedagogical issues posed by negative numbers at the pre-college level?
Suggestions for references, and comments of all kinds, are welcome. I appreciate candid criticism from people who are sympathetic to my aims but think I've fallen short of them. My prose style is designed to make the material accessible, but my breeziness should not be interpreted as indicating indifference to scholarly correctness.
Please leave your feedback at https://mathenchant.wordpress.com?p=1196&shareadraft=57d777dd7094f keeping in mind that the comments of EVERYONE on math-fun will get funneled through the same email conduit; if you want me to know who you are, please sign your comments!
If you sign your comments I'll assume (unless you indicate otherwise) that you don't mind my acknowledging your contribution.
Title: Going Negative Beginning: Minus times minus equals plus / The reason for this we will not discuss. --- W. H. Auden, recalling a popular verse from his school days Ever tried mixing together your two least favorite foods? I suspect you haven't. Nobody mixes two noxious ingredients and expects the results to be tasty... Read more: https://mathenchant.wordpress.com?p=1196&shareadraft=57d777dd7094f
Thanks,
Jim Propp _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
I think the "practical" justification is best: Just like defining empty sums as 0 and empty products as 1, and 0! = 1, and 0^0 = 1 sometimes, and primes starting at 2: There's arguable logic for several choices, and mathematicians have settled on the most convenient choice, that leads to the simplest formulas and theorems. There's a modest physics justification, based on charge: There are positive and negative charges, and the electric force between two charged particles behaves like our usual sign convention: flipping the sign of either charge flips the force, and flipping the signs of both charges flips the force back. Note that set theory has some problems, because union and intersection are idempotent, and order of operations sometimes makes a difference: Interpreting A+B as (A union B) and A*B as (A intersect B), the distributive law works; in fact union distributes over both union and intersection, as does intersection. But things get messy when we bring in subtraction. If we follow the usual convention, A - B means A - (A intersect B), or A intersect (B complement). But then (A - B) + C is >= (A + C) - B; and the distributive law is a mess. Is there even a formula for (A-B)*(C-D)? Rich ------- Quoting James Propp <jamespropp@gmail.com>:
I started writing a new draft titled "Going Negative" and would love to get your feedback. I plan on publishing it on the 17th and will continue to tinker with it between now and the weekend.
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James Propp -
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Tom Karzes